40 lines
2.2 KiB
Text
40 lines
2.2 KiB
Text
/-
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Copyright (c) 2017 Kuen-Bang Hou (Favonia).
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Kuen-Bang Hou (Favonia)
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-/
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import .homology .sphere ..susp_product
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open eq pointed group algebra circle sphere nat equiv susp
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function sphere homology int lift prod smash
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namespace homology
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section
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parameter (theory : ordinary_homology_theory)
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open ordinary_homology_theory
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theorem Hptorus : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m ×* psphere m)) ≃g
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HH theory n (plift (psphere m)) ×g (HH theory n (plift (psphere m)) ×g HH theory n (plift (psphere (m + m)))) := λ n m,
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calc HH theory n (plift (psphere m ×* psphere m))
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≃g HH theory (n + 1) (plift (⅀ (psphere m ×* psphere m))) : by exact (Hplift_psusp theory n (psphere m ×* psphere m))⁻¹ᵍ
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... ≃g HH theory (n + 1) (plift (⅀ (psphere m) ∨ (⅀ (psphere m) ∨ ⅀ (psphere m ∧ psphere m))))
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: by exact Hplift_isomorphism theory (n + 1) (susp_product (psphere m) (psphere m))
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... ≃g HH theory (n + 1) (plift (⅀ (psphere m))) ×g HH theory (n + 1) (plift (⅀ (psphere m) ∨ (⅀ (psphere m ∧ psphere m))))
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: by exact Hplift_pwedge theory (n + 1) (⅀ (psphere m)) (⅀ (psphere m) ∨ (⅀ (psphere m ∧ psphere m)))
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... ≃g HH theory n (plift (psphere m)) ×g (HH theory n (plift (psphere m)) ×g HH theory n (plift (psphere (m + m))))
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: by exact product_isomorphism (Hplift_psusp theory n (psphere m))
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(calc
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HH theory (n + 1) (plift (⅀ (psphere m) ∨ (⅀ (psphere m ∧ psphere m))))
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≃g HH theory (n + 1) (plift (⅀ (psphere m))) ×g HH theory (n + 1) (plift (⅀ (psphere m ∧ psphere m)))
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: by exact Hplift_pwedge theory (n + 1) (⅀ (psphere m)) (⅀ (psphere m ∧ psphere m))
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... ≃g HH theory n (plift (psphere m)) ×g HH theory n (plift (psphere (m + m)))
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: by exact product_isomorphism (Hplift_psusp theory n (psphere m))
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(Hplift_psusp theory n (psphere m ∧ psphere m) ⬝g Hplift_isomorphism theory n (sphere_smash_sphere m m)))
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end
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end homology
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